# On the Limitations of Fractal Dimension as a Measure of Generalization

arXiv: https://arxiv.org/abs/2406.02234 · HTML: https://mlsanity.com/paper/2406.02234/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 48 references that are in the map.
- First posted on arXiv 4 Jun 2024.

## Builds on (most cited first, 19 of 19)

- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Asymmetric Valleys: Beyond Sharp and Flat Local Minima](https://mlsanity.com/paper/1902.00744/) (2019)
- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networks](https://mlsanity.com/paper/2006.09313/) (2020)
- [giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration](https://mlsanity.com/paper/2004.02551/) (2020)
- [Bad Global Minima Exist and SGD Can Reach Them](https://mlsanity.com/paper/1906.02613/) (2019)
- [On the Heavy-Tailed Theory of Stochastic Gradient Descent for Deep Neural Networks](https://mlsanity.com/paper/1912.00018/) (2019)
- [Generalization bounds for deep learning](https://mlsanity.com/paper/2012.04115/) (2020)
- [Fractal Structure and Generalization Properties of Stochastic Optimization Algorithms](https://mlsanity.com/paper/2106.04881/) (2021)
- [Local permutation tests for conditional independence](https://mlsanity.com/paper/2112.11666/) (2021)
- [Persistent Homology and the Upper Box Dimension](https://mlsanity.com/paper/1802.00533/) (2018)
- [Generalization Bounds with Data-dependent Fractal Dimensions](https://mlsanity.com/paper/2302.02766/) (2023)
- [Distributed computation of persistent homology](https://mlsanity.com/paper/1310.0710/) (2013)
- [Discrete Morse Sandwich: Fast Computation of Persistence Diagrams for Scalar Data -- An Algorithm and A Benchmark](https://mlsanity.com/paper/2206.13932/) (2022)

## Built on by (most cited first, 5 of 5)

- [T-REGS: Minimum Spanning Tree Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2510.23484/) (2025)
- [Topological Signatures of Grokking](https://mlsanity.com/paper/2605.06352/) (2026)
- [IdEst: Assessing Self-Supervised Learning Representations via Intrinsic Dimension](https://mlsanity.com/paper/2606.03338/) (2026)
- [Stability, Complexity and Data-Dependent Worst-Case Generalization Bounds](https://mlsanity.com/paper/2507.06775/) (2025)
- [On the Stability and Generalization of First-order Bilevel Minimax Optimization](https://mlsanity.com/paper/2604.20115/) (2026)

## Abstract (arXiv)

Bounding and predicting the generalization gap of overparameterized neural networks remains a central open problem in theoretical machine learning. There is a recent and growing body of literature that proposes the framework of fractals to model optimization trajectories of neural networks, motivating generalization bounds and measures based on the fractal dimension of the trajectory. Notably, the persistent homology dimension has been proposed to correlate with the generalization gap. This paper performs an empirical evaluation of these persistent homology-based generalization measures, with an in-depth statistical analysis. Our study reveals confounding effects in the observed correlation between generalization and topological measures due to the variation of hyperparameters. We also observe that fractal dimension fails to predict generalization of models trained from poor initializations. We lastly reveal the intriguing manifestation of model-wise double descent in these topological generalization measures. Our work forms a basis for a deeper investigation of the causal relationships between fractal geometry, topological data analysis, and neural network optimization.